Noninteracting Walks Research Articles

Floquet Anderson localization of two interacting discrete time quantum walks

We study the interplay of two interacting discrete time quantum walks in the presence of disorder. Each walk is described by a Floquet unitary map defined on a chain of two-level systems. Strong disorder induces a novel Anderson localization phase with a gapless Floquet spectrum and one unique localization length $\xi_1$ for all eigenstates for noninteracting walks. We add a local contact interaction which is parametrized by a phase shift $\gamma$. A wave packet is spreading subdiffusively beyond the bounds set by $\xi_1$ and saturates at a new length scale $\xi_2 \gg \xi_1$. In particular we find $\xi_2 \sim \xi_1^{1.2}$ for $\gamma=\pi$. We observe a nontrivial dependence of $\xi_2$ on $\gamma$, with a maximum value observed for $\gamma$-values which are shifted away from the expected strongest interaction case $\gamma=\pi$. The novel Anderson localization regime violates single parameter scaling for both interacting and noninteracting walks.

Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents**All series data generated for this paper are available with the preprint at arXiv:1911.05852.

We investigate, by series methods, the behaviour of interacting self-avoiding walks (ISAWs) on the honeycomb lattice and on the square lattice. This is the first such investigation of ISAWs on the honeycomb lattice. We have generated data for ISAWs up to 75 steps on this lattice, and 55 steps on the square lattice. For the hexagonal lattice we find the θ-point to be at uc = 2.767 ± 0.002. The honeycomb lattice is unique among the regular two-dimensional lattices in that the exact growth constant is known for non-interacting walks, and is (Duminil-Copin H and Smirnov S 2014 Ann. Math. 175 1653–65), while for half-plane walks interacting with a surface, the critical fugacity, again for the honeycomb lattice, is (Beaton N R et al 2014 Commun. Math. Phys. 326 727–54). We could not help but notice that We discuss the difficulties of trying to prove, or disprove, this possibility. For square lattice ISAWs we find uc = 1.9474 ± 0.001, which is consistent with the best Monte Carlo analysis. We also study bridges and terminally-attached walks (TAWs) on the square lattice at the θ-point. We estimate the exponents to be γb = 0.00 ± 0.03, and γ1 = 0.55 ± 0.03 respectively. The latter result is consistent with the prediction (Duplantier B and Saleur H 1987 Phys. Rev. Lett. 59 539–42; Seno F and Stella A L 1988 Europhys. Lett. 7 605–10; Stella A L et al 1993 J. Stat. Phys. 73 21–46) , albeit for a modified version of the problem, while the former estimate is predicted in [Duplantier B and Guttmann A J 2019 Statistical mechanics of confined polymer networks (in preparation)] to be zero.

Noninteracting multiparticle quantum random walks applied to the graph isomorphism problem for strongly regular graphs

We investigate the quantum dynamics of particles on graphs ("quantum random walks"), with the aim of developing quantum algorithms for determining if two graphs are isomorphic (related to each other by a relabeling of vertices). We focus on quantum random walks of multiple non-interacting particles on strongly regular graphs (SRGs), a class of graphs with high symmetry that is known to have pairs of graphs that are hard to distinguish. Previous work has already demonstrated analytically that two-particle non-interacting quantum walks cannot distinguish non-isomorphic SRGs of the same family. Here, we demonstrate numerically that three-particle non-interacting quantum walks have significant, but not universal, distinguishing power for pairs of SRGs, proving a fundamental difference between the distinguishing power of two-particle and three-particle non-interacting walks. We analytically show why this distinguishing power is possible, whereas it is forbidden for two-particle non-interacting walks. Based on sampling of SRGs with up to 64 vertices, we find no difference in the distinguishing power of bosonic and fermionic walks. In addition, we find that the four-fermion non-interacting walk has greater distinguishing power than the three-particle walks on SRGs, showing that increasing particle number increases distinguishing power. However, we also analytically show that no non-interacting walk with a fixed number of particles can distinguish all SRGs, thus demonstrating a potential fundamental difference between the distinguishing power of interacting and noninteracting walks.